I'm trying to explain Gauss' Law to a bunch of high school kids. They wanna know why the total electric flux through a closed surface does not depend on the shape and volume of the closed surface and the orientation of the enclosed charges. I know the math, but conceptually, I'm at a loss for words. :(
Can you get them to accept the concept and basic properties of field lines? If you assume that field lines always start and end on charges, and that the flux through a surface is proportional to the number of fields lines that "pierce" it, then Gauss's Law follows. You might also try a fluid analogy. A positive charge is like the end of a hose with water flowing out of it, and a negative charge is like a drain hole where water flows in. The flux through a surface is the number of gallons (or liters) per second.
If it would depend on the shape (without additional charges somewhere), you could find a volume without charges, but with non-zero total flux. You could then divide that volume into smaller parts, and you would always get at least one part with non-zero total flux, but without charges. Maxwell's laws allow to derive ##div(E)=\rho## plus prefactors. In particular, for a very small volume, E is nearly constant. It is hard to imagine a very small volume with zero divergence inside, but non-zero total flux on its surface.